On Singer's conjecture for the fourth algebraic transfer in certain generic degrees

TL;DR

This paper proves Singer's conjecture for the algebraic transfer of rank four in specific generic degrees, confirming its injectivity in these cases and completing the proof for rank four, while noting it fails for rank six.

Contribution

The paper establishes Singer's conjecture for rank four in certain degrees, using novel algorithms, and completes the proof for this rank, advancing understanding of the algebraic transfer.

Findings

Singer's conjecture holds for rank four in specified degrees.

The authors' algorithms verify the results directly.

Singer's conjecture does not hold for rank six.

Abstract

Let $A$ be the Steenrod algebra over the finite field $k := \mathbb Z_2$ and $G(q)$ be the general linear group of rank $q$ over $k.$ A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, ${\rm Ext}^{q, *}_A(k, k),$ for all homological degrees $q \geq 0.$ The Singer algebraic transfer of rank $q,$ formulated by William Singer in 1989, serves as a valuable method for the description of such Ext groups. This transfer maps from the coinvariants of a certain representation of $G(q)$ to ${\rm Ext}^{q, *}_A(k, k).$ Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all $q\geq 4.$ This paper establishes Singer's conjecture for rank four in the generic degrees $n = 2^{s+t+1} +2^{s+1} - 3$ whenever $t\neq 3$ and $s\geq 1,$ and $n = 2^{s+t} + 2^{s} - 2$ whenever $t\neq 2,\, 3,\, 4$ and $s\geq 1.$ In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. All the obtained results can be verified directly by using the program suite of our novel algorithms presented in [17, 18, 19, 20]. We note that although Singer's conjecture still holds for the case of rank 4, it no longer holds for rank 6, as announced in our most recent work [20].